So, my impression on looking at the photos is that the Garantia is of a current styling, the bottle looks molded (not blown) per the shoulder seam in photo 3, the labels look like a mishmash of modern and antiqued, and the capsule is similar to what Niepoort has used recently. I think i recall opening up even slightly older niepoorts and the capsule was yellow, and back in '31 my guess is it would have been waxed instead.

The label says Bottled 1938 and Decanted 1979. This is a 1931 Garrafeira - aged in wood for 7 years then another 41 years in Demijon before being "Decanted" into this bottle. The inconsistent apparent ages of the label, seal and selo can easily be explained if it was released from the cellar some time after 1979. That would also explain why the selo is on the outside of the capsule.

That definitely explains the modern trappings on a a bottle labeled 1931, but is this a 1931 Niepoort "VP", or something else? What effect has Demijon>Bottle had (based on our tastings)? And has anyone seen this before from Niepoort?

You can try looking for a consigner (Benchmark) or for a wine auctioneer (zachys, acker merrall condit), or you can look for one of the "sell your wine" online sites/auctions. In any case, you'll be asked for details on how the wine was kept.

From the label, this appears to be a 1931 Niepoort Garrafeira port - note this is neither a "vintage port" nor a "tawny"/"colheita", but a separate relatively uncommon style of port created by the process described by DRT.

The label and bottle appear genuine, of the expected type. The capsule is the same colour as those I have seen on Ni31g previously, though I think I have only seen plain red capsules. The selo seems suprisingly clean to me given the label, but this is something I have seen before on another similar bottle so perhaps not an issue; though I would note that the selo on previous bottles has always been under the capsule rather than over it - which is my only warning flag from the images - though I also note that the auction you linked on Catawiki has a very similar looking bottle to yours in every respect.

Regarding value, this will depend on the bottle provenance, where it has come from and how it has been stored, if known. Can you comment on this last?

In broad terms, Garrafeiras should be treated and stored like Vintage Ports (or red wines). Unfortunately, stored in an air conditioned home not in a cellar or fridge is not proper storage conditions. It's not horrible, but it's not what I would expect for an expensive bottle of wine. So to me, this bottle would not be worth what a properly stored bottle might fetch.

Also note that a private sale will rarely fetch the prices that you see on the internet. In the US you can often approach 70% of internet prices; in the UK it's more like 50%. So while a properly stored bottle like this might sell for $1000 via an internet retailer, it would likely only sell for $700 (if you're lucky) in the US or $500 in the UK.

Factor in the storage conditions and I would estimate $350 to $450 in the US for a private sale. Perhaps $300 in the UK.

The states are complicated as every state has different laws and shipping wine as a private person is even more difficult (as in if you ship it illegally they confiscate it and destroy it if they find it). Add in no shipping until at least October too due to heat too. Best best is an auction house or sell to someone in your state you can hand deliver to. Or wait until winter and then sell it (probably the best bet).